where \(K(P^\star , P_\theta ) = \int \log (dP^\star / dP_\theta ) \, dP^\star \)  is the Kullback–Leibler divergence of \(P_\theta \)  from \(P^\star \) .
In general, \(\theta ^\dagger \)  does not have a real-life interpretation but, in some cases, certain features of \(P^\star \)  can be identified based on a misspecified model. For example, if \({P}\)  is an exponential family, then the mean function of the exponential family model, evaluated at \(\theta ^\dagger \) , equals the mean of \(P^\star \)  [1]}. Another similar case is considered in Section . The second challenge is that, even in the case where \(\theta ^\dagger \)  has a (limited) real-life interpretation, misspecification can still negatively impact posterior inferences. [2]} established a version of the Bernstein–von Mises theorem under model misspecification which states that, under certain regularity conditions, the posterior \(\Pi _n\)  will be approximately normal, with mean equal to the maximum likelihood (or M-) estimator \(\hat{\theta }\)  and covariance matrix \(V_{\theta ^\dagger }^{-1}\) , where
\( V_{\theta ^\dagger } = \int \Bigl ( \frac{\partial ^2 \log p_\theta }{\partial \theta \partial \theta ^\top } \Bigr |_{\theta =\theta ^\dagger } \Bigr ) \, dP^\star , \) 
